document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1158813>Question 1158813</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A group of 2519 students were surveyed about the courses they were taking at their college with the following results:<BR>\n" );
document.write( "1390 students said they were taking History.<BR>\n" );
document.write( "1181 students said they were taking English.<BR>\n" );
document.write( "1318 students said they were taking Science.<BR>\n" );
document.write( "666 students said they were taking English and Science.<BR>\n" );
document.write( "613 students said they were taking History and English.<BR>\n" );
document.write( "744 students said they were taking History and Science.<BR>\n" );
document.write( "359 students said they were taking all three courses.\r<BR>\n" );
document.write( "\n" );
document.write( "a) Fill in the following Venn Diagram with the cardinality of each region.<BR>\n" );
document.write( "History = I,II,IV,V<BR>\n" );
document.write( "English = II,III,V,VI<BR>\n" );
document.write( "Science = IV, V,VI,VII<BR>\n" );
document.write( "students took none of the courses = 294<BR>\n" );
document.write( "I. = 392<BR>\n" );
document.write( "II. = 254<BR>\n" );
document.write( "III. = 261<BR>\n" );
document.write( "IV. = 385<BR>\n" );
document.write( "V. = 359<BR>\n" );
document.write( "VI. = 307<BR>\n" );
document.write( "VII. = 267<BR>\n" );
document.write( "VIII. = 294\r<BR>\n" );
document.write( "\n" );
document.write( "a) How many students took History or didn't take English?\r<BR>\n" );
document.write( "\n" );
document.write( "b) How many students took History & English or took English & Science?\r<BR>\n" );
document.write( "\n" );
document.write( "c) How many students took History or Science, but not English?\r<BR>\n" );
document.write( "\n" );
document.write( "d) How many students took History, English, or Science?\r<BR>\n" );
document.write( "\n" );
document.write( "e) How many students took none of the courses?\r<BR>\n" );
document.write( "\n" );
document.write( "f) How many students took English and Science, but not History? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854642 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5399)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854642\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The Venn diagram to be filled was not included in the question, so I will draw one.<BR>\n" );
document.write( "We need 3 circles to \"hold in\" all the students that were taking one of the 3 courses mentioned.<BR>\n" );
document.write( "The circles have to intersect, so that there is a region that is included in all 3 circles at the same time.<BR>\n" );
document.write( "I will add color to the diagram to more easily identify the circles for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28History%29 ALIGN=MIDDLE  ALT=\"blue%28History%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28English%29 ALIGN=MIDDLE  ALT=\"red%28English%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28Science%29 ALIGN=MIDDLE  ALT=\"green%28Science%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I will outline the circles in the corresponding color, and will arrange them in that order going left to right and top to bottom.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-4%2C4%2C-4%2C4%2Cgreen%28circle%280%2C-1.3%2C2%29%29%2C%0D%0Ablue%28circle%28-1.4%2C1%2C2%29%29%2Cred%28circle%281.4%2C1%2C2%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "a) Fill in the Venn Diagram with the cardinality (number of students) of each region.<BR>\n" );
document.write( "The diagram shows 7 regions that are part of at least one circle.<BR>\n" );
document.write( "An additional 8th region, the part not included in any circle, would \"hold\" the students who are not taking any of the courses mentioned.<BR>\n" );
document.write( "Presumably. there was a blank Venn diagram given with those 8 regions labeled with the Roman numerals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I ALIGN=MIDDLE  ALT=\"I\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=II ALIGN=MIDDLE  ALT=\"II\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=III ALIGN=MIDDLE  ALT=\"III\"   DISABLED_style= \"max-width:90%; height:auto;\" >, etc,<BR>\n" );
document.write( "but I assume the number of students corresponding to each region was not given, but found by the student, so I will not use that data.<BR>\n" );
document.write( "The almost triangular region in the center of the diagram \"holds\" the students who are taking all 3 of the courses mentioned.<BR>\n" );
document.write( "We are given the number of students there, so I will enter that number: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=359 ALIGN=MIDDLE  ALT=\"359\"   DISABLED_style= \"max-width:90%; height:auto;\" > in that region of the diagram.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-4%2C4%2C-4%2C4%2Cgreen%28circle%280%2C-1.3%2C2%29%29%2C%0D%0Ablue%28circle%28-1.4%2C1%2C2%29%29%2Cred%28circle%281.4%2C1%2C2%29%29%2C%0D%0Alocate%28-0.2%2C0.3%2C359%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Numbers of students taking exactly 2 of the subjects listed:<BR>\n" );
document.write( "The number of students in the lense-shaped circle overlaps are given as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=666 ALIGN=MIDDLE  ALT=\"666\"   DISABLED_style= \"max-width:90%; height:auto;\" > students that said they were taking English and Science,			<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=613 ALIGN=MIDDLE  ALT=\"613\"   DISABLED_style= \"max-width:90%; height:auto;\" > students that said they were taking History and English, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=744 ALIGN=MIDDLE  ALT=\"744\"   DISABLED_style= \"max-width:90%; height:auto;\" > students that said they were taking History and Science.<BR>\n" );
document.write( "All of those could also have been taking the other listed subject.<BR>\n" );
document.write( "Subtracting from each of those numbers the number of student who were taking all 3 courses, we can calculate the number of students who were taking only 2 of the subjects:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=666-359=highlight%28307%29 ALIGN=MIDDLE  ALT=\"666-359=highlight%28307%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > students that were taking English and Science, but not History, 		 	<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=613-359=highlight%28254%29 ALIGN=MIDDLE  ALT=\"613-359=highlight%28254%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > students that were taking History and English, but not <BR>\n" );
document.write( "Science, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=744-359=highlight%28385%29 ALIGN=MIDDLE  ALT=\"744-359=highlight%28385%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > students that were taking History and Science, but not English.<BR>\n" );
document.write( "Numbers of students taking only 1 of the subjects listed can be calculated based on the results above, or independently.<BR>\n" );
document.write( "The part of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28History%29 ALIGN=MIDDLE  ALT=\"blue%28History%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > circle that is not in any other circle represents the number of students that were taking History, but not another of the listed subjects.<BR>\n" );
document.write( "That number can be calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1390-359-254-385=highlight%28392%29 ALIGN=MIDDLE  ALT=\"1390-359-254-385=highlight%28392%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , found by subtracting from the number that said they were taking history, the numbers that were history and just one other subject, and the number that were taking all 3 subjects.<BR>\n" );
document.write( "It can also be calculated without relying on the previous results as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1390-613-744%2B359=highlight%28392%29 ALIGN=MIDDLE  ALT=\"1390-613-744%2B359=highlight%28392%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "found by subtracting from the number that said they were taking history, the numbers that were taking a second subject,<BR>\n" );
document.write( "and then adding the number of those subtracted twice because they were taking all 3  subjects.<BR>\n" );
document.write( "The two results should be the same, an could be used as verification.<BR>\n" );
document.write( "The other regions that are part of only one circle can be calculated similarly.<BR>\n" );
document.write( "The part of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28English%29 ALIGN=MIDDLE  ALT=\"red%28English%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > circle that is not in any other circle represents the number of students that were taking English, but not another of the listed subjects.<BR>\n" );
document.write( "That number can be calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1181-359-307-254=highlight%28261%29 ALIGN=MIDDLE  ALT=\"1181-359-307-254=highlight%28261%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1181-666-613%2B359=highlight%28261%29 ALIGN=MIDDLE  ALT=\"1181-666-613%2B359=highlight%28261%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The part of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28Science%29 ALIGN=MIDDLE  ALT=\"green%28Science%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > circle that is not in any other circle represents the number of students that were taking science, but not another of the listed subjects.<BR>\n" );
document.write( "That number can be calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1318-359-307-385=highlight%28267%29 ALIGN=MIDDLE  ALT=\"1318-359-307-385=highlight%28267%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1318-666-744%2B359=highlight%28267%29 ALIGN=MIDDLE  ALT=\"1318-666-744%2B359=highlight%28267%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "Number of students who took at least one of the courses listed:<BR>\n" );
document.write( "Calculated as the sum of the numbers for all 7 regions in side one or more circles, that number is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=359%2B307%2B254%2B385%2B392%2B261%2B267=2225 ALIGN=MIDDLE  ALT=\"359%2B307%2B254%2B385%2B392%2B261%2B267=2225\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Number of students who did not take any of the courses listed:<BR>\n" );
document.write( "That number can be calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2519-2225=highlight%28294%29 ALIGN=MIDDLE  ALT=\"2519-2225=highlight%28294%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Entering the cardinalities (numbers of students) for those regions and the roman numerals for them as given by the person posting the question, we have:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-4%2C4%2C-4%2C4%2Cgreen%28circle%280%2C-1.3%2C2%29%29%2C%0D%0Ablue%28circle%28-1.4%2C1%2C2%29%29%2Cred%28circle%281.4%2C1%2C2%29%29%2C%0D%0Alocate%28-0.05%2C0.6%2CV%29%2Clocate%28-0.2%2C0.3%2C359%29%2C%0D%0Alocate%28-0.13%2C1.4%2CII%29%2Clocate%28-0.2%2C1.1%2C254%29%2C%0D%0Alocate%28-1.6%2C1.7%2Cblue%28I%29%29%2Clocate%28-1.75%2C1.4%2Cblue%28392%29%29%2C%0D%0Alocate%281.4%2C1.7%2Cred%28III%29%29%2Clocate%281.4%2C1.4%2Cred%28261%29%29%2C%0D%0Alocate%28-0.25%2C-1.3%2Cgreen%28VII%29%29%2Clocate%28-0.25%2C-1.6%2Cgreen%28267%29%29%2C%0D%0Alocate%28-1%2C0%2CIV%29%2Clocate%28-1%2C-0.3%2C385%29%2C%0D%0Alocate%280.8%2C0%2CVI%29%2Clocate%280.8%2C-0.3%2C307%29%2C%0D%0Alocate%282%2C-2.7%2CVIII%29%2Clocate%282%2C-3%2C294%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "a) How many students took History or didn't take English?<BR>\n" );
document.write( "The students who took History are represented in regions I, II, IV, and V.<BR>\n" );
document.write( "The students who didn't take English are in regions I,IV, VII, and VIII.<BR>\n" );
document.write( "All of those regions are involved because it says or, so it's a union of both sets.<BR>\n" );
document.write( "The union of those two sets includes regions I,II, IV, V, VII, and VIII.<BR>\n" );
document.write( "That leaves out only regions III, and VI, so the number can be calculated subtracting the number in those 2 regions from the total as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2519-261-307=highlight%281951%29 ALIGN=MIDDLE  ALT=\"2519-261-307=highlight%281951%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "b) How many students took History & English or took English & Science?<BR>\n" );
document.write( "The students who took History & English are represented in regions II and V.<BR>\n" );
document.write( "The students who took English & Science are represented in regions V and VI.<BR>\n" );
document.write( "The answer is again in the union of those 2 sets: regions II, V, and VI:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=254%2B359%2B307=highlight%28920%29 ALIGN=MIDDLE  ALT=\"254%2B359%2B307=highlight%28920%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "c) How many students took History or Science, but not English?<BR>\n" );
document.write( "That's those who took one of the courses listed but did not take English.<BR>\n" );
document.write( "I could calculate their number as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2519-294-1181=highlight%281044%29 ALIGN=MIDDLE  ALT=\"2519-294-1181=highlight%281044%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "I could also calculate it as the number in regions I, IV, and VII, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=392%2B385%2B267=highlight%281044%29 ALIGN=MIDDLE  ALT=\"392%2B385%2B267=highlight%281044%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "d) How many students took History, English, or Science?<BR>\n" );
document.write( "That's those who took at least one of the courses listed, so it can be calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2519-294=highlight%282225%29 ALIGN=MIDDLE  ALT=\"2519-294=highlight%282225%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "e) How many students took none of the courses?<BR>\n" );
document.write( "That's the number in region VIII: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28294%29 ALIGN=MIDDLE  ALT=\"highlight%28294%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "f) How many students took English and Science, but not History?<BR>\n" );
document.write( "The students who took English and sciences are represented by regions V and VI.<BR>\n" );
document.write( "The students in region V had taken also History, so they are not included in the answer.<BR>\n" );
document.write( "So, the answer is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28307%29 ALIGN=MIDDLE  ALT=\"highlight%28307%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > students in region VI. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );